It is used throughout real analysis, in particular to define Lebesgue integration.

Such generalisations are often formulated in terms of Lebesgue integration.

Finally we can define the conditional mutual information via Lebesgue integration:

This can be thought of as a Bayesian's way to numerically implement Lebesgue integration.

In his textbooks of real analysis, he introduced a definition of the integral analogous to Lebesgue integration.

This constructive measure theory provides the basis for computable analogues for Lebesgue integration.

The Riesz-Fischer theorem in Lebesgue integration is named in his honour.

Henri Lebesgue introduces the theory of Lebesgue integration.

This approximation of by simple functions (which are easily integrable) allows us to define an integral itself; see the article on Lebesgue integration for more details.

Its power and utility are two of the primary theoretical advantages of Lebesgue integration over Riemann integration.

During this time he taught a course on the then cutting edge theory of Lebesgue integration, one of the first such courses offered outside of France.

For many authors, the Riesz-Fischer theorem refers to the fact that the L spaces from Lebesgue integration theory are complete.

The most common notion of integration is based on the abstract mathematical theory known as Lebesgue integration, developed by Henri Lebesgue.

The statement for the integral form of the remainder is more advanced than the previous ones, and requires understanding of Lebesgue integration theory for the full generality.

The first of these, unrelated to his development of Lebesgue integration, dealt with the extension of Baire's theorem to functions of two variables.

Later he proved some foundational results concerning Lebesgue integration, including a statement that even today appears in many measure theory textbooks as "Beppo Levi's lemma".

This is a non-technical treatment from a historical point of view; see the article Lebesgue integration for a technical treatment from a mathematical point of view.

Lebesgue integration is a mathematical construction that extends the integral to a larger class of functions; it also extends the domains on which these functions can be defined.

To illustrate some of the proof techniques used in Lebesgue integration theory, we sketch a proof of the above mentioned Lebesgue monotone convergence theorem.

A technical issue in Lebesgue integration is that the domain of integration is defined as a "set" (a subset of a measure space), with no notion of orientation.

In measure theory, Lebesgue's dominated convergence theorem provides sufficient conditions under which two limit processes commute, namely Lebesgue integration and almost everywhere convergence of a sequence of functions.

Using the general theory of Lebesgue integration, one can then define an integral for all Borel measurable functions f on G. This integral is called the Haar integral.

A useful characterisation of Lebesgue measurable functions is that f is measurable if and only if mid is integrable for all non-negative Lebesgue integration functions g.

In measure-theoretic analysis and related branches of mathematics, Lebesgue-Stieltjes integration generalizes Riemann-Stieltjes and Lebesgue integration, preserving the many advantages of the former in a more general measure-theoretic framework.

Lebesgue integration has the property that every bounded function defined over a bounded interval with a Riemann integral also has a Lebesgue integral, and for those functions the two integrals agree.